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Inference for covariate adjusted regression via varying coefficient models. (English) Zbl 1095.62045
Summary: We consider covariate adjusted regression (CAR), a regression method for situations where predictors and response are observed after being distorted by a multiplicative factor. The distorting factors are unknown functions of an observable covariate, where one specific distorting function is associated with each predictor or response. The dependence of both response and predictors on the same confounding covariate may alter the underlying regression relation between undistorted but unobserved predictors and response.
We consider a class of highly flexible adjustment methods for parameter estimation in the underlying regression model, which is the model of interest. Asymptotic normality of the estimates is obtained by establishing a connection to varying coefficient models. These distribution results combined with proposed consistent estimates of the asymptotic variance are used for the construction of asymptotic confidence intervals for the regression coefficients. The proposed approach is illustrated with data on serum creatinine, and finite sample properties of the proposed procedures are investigated through a simulation study.

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62J05 Linear regression; mixed models
62E20 Asymptotic distribution theory in statistics
65C05 Monte Carlo methods
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[1] Cai, Z., Fan, J. and Li, R. (2000). Efficient estimation and inferences for varying-coefficient models. J. Amer. Statist. Assoc. 95 888–902. JSTOR: · Zbl 0999.62052 · doi:10.2307/2669472 · links.jstor.org
[2] Chiang, C.-T., Rice, J. A. and Wu, C. O. (2001). Smoothing spline estimation for varying coefficient models with repeatedly measured dependent variables. J. Amer. Statist. Assoc. 96 605–619. JSTOR: · Zbl 1018.62034 · doi:10.1198/016214501753168280 · links.jstor.org
[3] Fan, J. and Zhang, J.-T (2000). Two-step estimation of functional linear models with applications to longitudinal data. J. R. Stat. Soc. Ser. B Methodol Stat. Methodol. 62 303–322. JSTOR: · Zbl 04558573 · doi:10.1111/1467-9868.00233 · links.jstor.org
[4] Härdle, W., Janssen, P. and Serfling, R. (1988). Strong uniform consistency rates for estimators of conditional functionals. Ann. Statist. 16 1428–1449. · Zbl 0672.62050 · doi:10.1214/aos/1176351047
[5] Hastie, T. and Tibshirani, R. (1993). Varying-coefficient models (with discussion). J. Roy. Statist. Soc. Ser. B 55 757–796. JSTOR: · Zbl 0796.62060 · links.jstor.org
[6] Hoover, D. R., Rice, J. A., Wu, C. O. and Yang, L.-P. (1998). Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika 85 809–822. JSTOR: · Zbl 0921.62045 · doi:10.1093/biomet/85.4.809 · www3.oup.co.uk
[7] Kaysen, G. A., Müller, H.-G., Young, B. S., Leng, X. and Chertow, G. M. (2004). The influence of patient- and facility-specific factors on nutritional status and survival in hemodialysis. J. Renal Nutrition 14 72–81.
[8] Lai, T. L., Robbins, H. and Wei, C. Z. (1979). Strong consistency of least-squares estimates in multiple regression. II. J. Multivariate Anal. 9 343–361. · Zbl 0416.62051 · doi:10.1016/0047-259X(79)90093-9
[9] McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. Ann. Probab. 2 620–628. · Zbl 0287.60025 · doi:10.1214/aop/1176996608
[10] Nadaraya, E. A. (1964). On estimating regression. Theory Probab. Appl. 9 141–142. · Zbl 0136.40902
[11] Şentürk, D. and Müller, H.-G. (2005). Covariate-adjusted regression. Biometrika 92 75–89. · Zbl 1068.62082 · doi:10.1093/biomet/92.1.75
[12] Watson, G. S. (1964). Smooth regression analysis. Sankhyā Ser. A 26 359–372. · Zbl 0137.13002
[13] Wu, C. O. and Chiang, C.-T. (2000). Kernel smoothing on varying-coefficient models with longitudinal dependent variable. Statist. Sinica 10 433–456. · Zbl 0945.62047
[14] Wu, C. O. and Yu, K. F. (2002). Nonparametric varying-coefficient models for the analysis of longitudinal data. Internat. Statist. Rev. 70 373–393. · Zbl 1217.62046 · doi:10.2307/1403863
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